Finite difference method of first boundary problem for quasilinear parabolic systems (IV)

Abstract

More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system:

$$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$

whereu, ϕ, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and$u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$. For this problem, the convergence of iteration for the general difference schemes is proved.